Recognition of computationally constructed loci. (English) Zbl 1195.68105

Lebmeir, Peter; Richter-Gebert, Jürgen

Botana, Francisco (ed.) et al., Automated deduction in geometry. 6th international workshop, ADG 2006, Pontevedra, Spain, August 31–September 2, 2006. Revised papers. Berlin: Springer (ISBN 978-3-540-77355-9/pbk). Lecture Notes in Computer Science 4869. Lecture Notes in Artificial Intelligence, 52-67 (2007).

Summary: We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.
For the entire collection see [Zbl 1132.68006].

MSC:

68U05	Computer graphics; computational geometry (digital and algorithmic aspects)
68T15	Theorem proving (deduction, resolution, etc.) (MSC2010)

Software:

Cinderella